Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability ${E}_{p},$ the percolation probability $P,$ and the probability ${W}_{n}$ for the appearance of $n$ percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions $L.$ Using the HMCSM, we calculate the existence probability ${E}_{p}$ and the percolation probability $P$ for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of ${E}_{p}$ and $P$ in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that ${E}_{p}$ and $P$ have universal finite-size scaling functions. This implies that the critical ${E}_{p}$ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that ${W}_{n}$ for site and bond percolation on sc lattices have universal finite-size scaling functions.Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability ${E}_{p},$ the percolation probability $P,$ and the probability ${W}_{n}$ for the appearance of $n$ percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions $L.$ Using the HMCSM, we calculate the existence probability ${E}_{p}$ and the percolation probability $P$ for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of ${E}_{p}$ and $P$ in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that ${E}_{p}$ and $P$ have universal finite-size scaling functions. This implies that the critical ${E}_{p}$ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that ${W}_{n}$ for site and bond percolation on sc lattices have universal finite-size scaling functions.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 1, 199850 citations

FRACTAL CLUSTERS AND SCALING IN THE ISING MODEL

J.‐L. Cambier, Michael Nauenberg

Elsevier eBooksJan 1, 19862 citations

Scaling and fractal dimension of Ising clusters at thed=2critical point

Attilio L. Stella, Carlo Vanderzande

Using formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager's critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d\ifmmode\bar\else\textasciimacron\fi{}=(187/96 and a percolative susceptibility exponent \ensuremath{\gamma}=(91/48, in good agreement with existing numerical estimates. This d\ifmmode\bar\else\textasciimacron\fi{} is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent ${\mathrm{y}}_{\mathrm{J}}$ =(13/24 controlling the crossover between clusters and droplets.Using formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager's critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d\ifmmode\bar\else\textasciimacron\fi{}=(187/96 and a percolative susceptibility exponent \ensuremath{\gamma}=(91/48, in good agreement with existing numerical estimates. This d\ifmmode\bar\else\textasciimacron\fi{} is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent ${\mathrm{y}}_{\mathrm{J}}$=(13/24 controlling the crossover between clusters and droplets.

Physical Review LettersMar 6, 1989111 citations

Cluster-size heterogeneity in the two-dimensional Ising model

Woo Seong Jo, Sudo Yi, Seung Ki Baek +1 more

We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents obtained via the finite-size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d(f) and the Fisher exponent τ for the cluster distribution. We also point out that strong finite-size effects exist due to the geometric nature of the cluster-size heterogeneity.We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents obtained via the finite-size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d(f) and the Fisher exponent τ for the cluster distribution. We also point out that strong finite-size effects exist due to the geometric nature of the cluster-size heterogeneity.

arXiv:1209.2568Physical Review ESep 10, 201211 citations

Dynamics of random fractals: large-scale simulations

Tsuneyoshi Nakayama

Physica A Statistical Mechanics and its ApplicationsDec 1, 199210 citations

{E}_{p},

the percolation probability

P,

and the probability

{W}_{n}

for the appearance of

n

percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions

L.

Using the HMCSM, we calculate the existence probability

{E}_{p}

and the percolation probability

P

for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of

{E}_{p}

and

P

in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that

{E}_{p}

and

P

have universal finite-size scaling functions. This implies that the critical

{E}_{p}

is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that

{W}_{n}

for site and bond percolation on sc lattices have universal finite-size scaling functions.

{\mathrm{y}}_{\mathrm{J}}

=(13/24 controlling the crossover between clusters and droplets.